滚动载荷作用下半空间表面裂纹的分析
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摘要
裂纹问题是断裂力学的重要研究课题,近三十年来,在三维断裂力学中,发展了一种求解三维裂纹问题的边界元数值方法。由于边界元法降低了问题的维数,大大的减少了计算量,所以用这种方法处理三维裂纹问题十分有效。根据边界元法的特征和接触问题的特点可知,边界元法比有限元法及其他数值方法更适合求解接触问题。目前接触问题已成为边界元法应用的重要方面。本文用边界元法求解了一般情况下的三维裂纹问题,并计算了在滚动载荷作用下半空间表面裂纹问题。本文首先在已有的两相材料空间界面裂纹的扰动应力场公式的基础上,导出了在半空间弹性体中,以裂纹面的位移间断为未知量,受任意载荷作用的三维平片裂纹的超奇异积分方程组。如果裂纹离自由面的距离很远,该方程组便可退化为无限空间弹性体中三维平片裂纹的超奇异积分方程组。在以上理论的基础上,使用有限部积分与常规边界元结合的方法,并利用位移(间断)的线性模型,将受一般载荷作用的三维平片裂纹的超奇异积分方程组离散为代数方程组,并对其中出现的各类奇异积分作了专门处理,为其建立了数值法,并给出了具体的计算公式。从而完成了超奇异积分方程组数值法的建立,这一方法现称之为有限部积分——边界元法。为了验证该方法的可靠性,本文对无限空间和半空间中矩形类、椭圆类和半椭圆类平片裂纹问题进行了计算。在半空间弹性体中所计算的应力强度因子的有效性基础上,对滚动载荷作用下的半椭圆表面裂纹和矩形表面裂纹问题进行了运算。分别讨论了在无液压和存在液压力的情况。对于a/b=2的半椭圆类表面裂纹和矩形表面裂纹的结果是文献上没有的,是本文获得的新结果。
The investigation of crack problems is very important in fracture mechanics. In the last thirty years, A boundary element method for three-dimensional crack problems have been developed. Due to the advantage of the boundary element method for reducing the dimension of the problems and calculation time, this method is very effective for dealing with three-dimensional crack problems. From the character of the boundary element method and that of contact problems, the boundary element method is more appropriate than other numerical methods for the contact problems. Now the contact problems have been important aspect in the application of the boundary element method. In the paper, the surface crack problems of semi-infinite solids in rolling load are discussed.On the basis of the stress formula of interfacial crack on the biomaterial solids, the paper derive hyper-singular integral equations with the unknown displacement discontinuities of a three-dimensional flat crack under arbitrary loads in semi-infinite elastic solids. If the crack is far from the free surface, the equations reduce to hyper-singular integral equations in infinite elastic solids.According to the above theory, the finite-part integral combined with boundary element method and linear models of the displacements (displacements discontinuities) are used to discretize the hyper-singular integral equations as a set of linear algebraic equations, where the singular integrals of various types are specially treated and the specific calculating formula are given. So far, the numerical techniques solving the hyper-singular integral equations are established, and these are called finite-part integral—boundary element method. To test the reliability of this method, rectangular crack elliptical crack and semi-elliptical crack in infinite space and in semi-infinite space are calculated.The result of stress intensity factor is very correct. Thus the problems of semi-elliptical surface crack and rectangular surface crack of semi-infinite solid in rolling load are discussed, including the fluid pressure case and no fluid pressure case. For the numerical result of semi-elliptical surface crack which a/b is equal to two and rectangular surface crack are inexistent in the literatures.
The investigation of crack problems is very important in fracture mechanics. In the last thirty years, A boundary element method for three-dimensional crack problems have been developed. Due to the advantage of the boundary element method for reducing the dimension of the problems and calculation time, this method is very effective for dealing with three-dimensional crack problems. From the character of the boundary element method and that of contact problems, the boundary element method is more appropriate than other numerical methods for the contact problems. Now the contact problems have been important aspect in the application of the boundary element method. In the paper, the surface crack problems of semi-infinite solids in rolling load are discussed.On the basis of the stress formula of interfacial crack on the biomaterial solids, the paper derive hyper-singular integral equations with the unknown displacement discontinuities of a three-dimensional flat crack under arbitrary loads in semi-infinite elastic solids. If the crack is far from the free surface, the equations reduce to hyper-singular integral equations in infinite elastic solids.According to the above theory, the finite-part integral combined with boundary element method and linear models of the displacements (displacements discontinuities) are used to discretize the hyper-singular integral equations as a set of linear algebraic equations, where the singular integrals of various types are specially treated and the specific calculating formula are given. So far, the numerical techniques solving the hyper-singular integral equations are established, and these are called finite-part integral—boundary element method. To test the reliability of this method, rectangular crack elliptical crack and semi-elliptical crack in infinite space and in semi-infinite space are calculated.The result of stress intensity factor is very correct. Thus the problems of semi-elliptical surface crack and rectangular surface crack of semi-infinite solid in rolling load are discussed, including the fluid pressure case and no fluid pressure case. For the numerical result of semi-elliptical surface crack which a/b is equal to two and rectangular surface crack are inexistent in the literatures.
引文
[1] 陈梦成,野田尚昭,三次元巽种接合弹性体中存在微小裂力埸(1,裂同一领域),机械研究,2003,55(5):567-578
[2]尹双增,断裂力学,北京,清华大学出版社,1992
[3]沈成康,断裂力学,上海,同济大学出版社,1996
[4] Kassir M. K.. Sih G. S., Three-dimensional stress distribution around an elliptical crack under arbitrary loading., J. Appl. Mech., 1963, 33 (E38): 601-644
[5] Kassir M. K., Sih G. S., Three-dimensional crack problems, Sih G C: Mechanics of Fracture II, Noordboff, leyden, 1975
[6]王银邦,非对称载荷作用的外部圆形裂纹问题,应用数学和力学,2001,22(1):9～15
[7]陈梦成,陈振建,汤任基,椭圆平片裂纹前沿应力强度因子解析解的超奇异积分方程方法.固体力学学报,1999,20(4):331-334
[8] Fleming J. R. and Suh N P., Mechanism of crack propagation in delamination wear, Wear, 1977, 44: 39-56
[9]Olzak M., Stupnicki J., Wijcik R., Investigation of crack propagation during contact by a finite element method, Wear, 1991, 146: 29-240
[10] Olzak M., Stupnicki J., Wijcik R., Numerical analysis of 3D cracks propagation in rail-wheel contact zone, The Netherlands: June 1992, 385-393
[11] Dubourg M C, Kalker J J,, Crack behavior under rolling contact fatigue, The Netherlands: June 1992, 373-384
[12] Bogdanski S., Brown M W., Modelling the three dimensional behavior of shallow rolling contact fatigue cracks in rails[A], Proceeding of the 5th International Conference on Contact Mechanics and Wear of Whell/Rail System[C], Japan: July 25 to 28, 2000, 9-16
[13] Kabo E., Ekberg A., Fatigue initiation in railway wheels on the influence of defects[A], Proceeding of the 5th International Conference on Contact Mechanics and Wear of Whell/Rail System[C], Japan: July 25 to 28, 2000, 17-22
[14] 杨庆生,孙艳丰,裂纹扩展过程的数值模拟,铁道学报,1997,19(5):116～120
[15] Cruse T. A., Numerical solutions in three dimensional elastostantics, Int. J. Solids Structures, 1969, 5: 1259-1274
[16]杜庆华,姚振汉,边界积分方程.边界元法的基本理论及其在弹性力学方面的若干工程应用,固体力学学报,1982,3(1):1-22
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[18] Du Qinghua, Yao Zhenhan, Cen Zhangzhi, Some applications of boundary element methods, boundary element-finite element coupling techniques in elastoplastic stress analysis, Acta Mechanica Solids Sinice, 1990, 3 (3): 327-340
[19]童丽萍,张晓萍,用边界元分析工程中的裂纹,郑州大学学报,1998,30(4):15—20
[20]刘建秀,乐金朝,弹性半空间中平片裂纹问题的超奇异积分方程方法,机械强度,1999,21(3):208-211
[21]陶伟明,郭乙木,三维有限裂纹体分析,固体力学学报,2001,22(3):256—262
[22] Cai Guozhong, Zhang Kangda, Stress intensity factors for semi-elliptical surface cracks in plates and cylindrical pressure vessels using the hybrid boundary element method, Engineering Fracture Mechanics, 1995, 52(6): 1035-1054
[23]周里群,李玉平,用边界元求解二维弹性接触中问题的改进方法,湘潭大学自然科学学报,1998,2:90-100
[24]徐玉辰,肖宏,用于边界元法的改进数值积分技术,燕山大学学报,2001,25(4):325-327
[25]佟维,考虑网格自动划分求敏度的边界元法,大连铁道学院学报,1997,18(1):10～16
[26]姚振汉,段小华,尹欣等,边界元法软件及其在工程与教学中的应用,重庆建筑大学学报,2000,22(6):84-87
[27] Noda Nao-aki, Miyoshi Sinsuke, Variation of stress intensity factor and crack opening displacement of semi-elliptical surface crack, International Journal of Fracture, 1996, 75: 19-48
[28] Murakami Y., Nwnat-Nasser S., Growth and stability of interacting surface flaws of arbitrary shape, Engineering Fracture Mechanics, 1983, 17 (3);193 — 210
[29] Murakami Y., Analysis of stress intensity factors of modes 1, II, and III for inclined surface cracks of arbitrary shape, Engineering Fracture Mechanics, 1985, 22(1): 101-114
[30]石田诚,鹤 秀登,野口博司,三次元裂问题高精度新解析法(第1报,解析基础理论无限体问题応用),日本机械学会论文集(A编),1993,59(561)1270-1278
[31]野口博司,石田 诚,鹤 秀登,三次元裂问题高精度新解析法(第2报,任意形状,表面·内部裂持半无限体问题～応用),日本机械学会论文集(A编),1993,59(561):1279-1286
[32]村上敬宜,兼田楨宏,八塚裕彦,接触下表面裂解析,日本机械学会论文集(C编),1985,51(467):1603-1610
[33]兼田楨宏,八塚裕彦,村上敬宜,接触下表面裂解析(裂接力作用),日本机械学会论文集(C编),1985,51(470):2618～2624
[34] Murakami Y., Kaneta M., Yatsuzuka Y., Analysis of surface crack propagation in
lubricated rolling contact, ASLE Transactions, 1985, 28 (1): 60-68
[35] Kaneta M., Yatsuzuka Y., Murakami Y., Mechanism of surface crack growth in lubricatedrolling/sliding spherical contact, Transactions of the ASME, 1986, 53: 354-360
[36]村上敬宜,一同题破坏力学応用,日本机械学会论文集(A编) 1993,59(558):283～290
[37] Ioakimidis N.I. , A natural approach to the introduction of finite-part integral into crack problems of three-dimensional elasticity, Engineering Fracture Mechanics, 1982, 16: 669~673
[38] Ioakimidis N.I. , Application of finite-part integral to the singular integral equations of crack problems in plane and three-dimensional elasticity, Acta Mechanics, 1982, 45: 31-47
[39]汤任基,断裂力学中的两类奇异积分方程,上海交通大学学报,1990,24:36-45
[40]幸筱流,陈梦成,汤任基,椭圆类平片裂纹二维有限部积分方程的多项式数值方法,南吕大学学报(工科类),1999,21(1):45-50
[41] Sohn G.H., Hong G.S., Application of singular integral equations to embedded planarcrack problems in finite body in boundary element 2, Brebbia C.A. Springer-Verlog, 1985,8: 57-67
[42] Hayashi K., Abe H., Stress intensity factors for a semi-elliptical crack in the surface of asemi-infinite solid, International Journal of Fracture, 1980, 16 (3): 275-285
[43] Fujimoto K., ItoH., YamamotoT., Effect of cracks on the contact pressure distribution.,Tribology Transactions. Vol.35, 1992, PP.683-695
[44] Keer L M., Bryant M D., A pitting model for rolling contact fatigue, Journal ofLubrication Technology, April 1983, 105: 198-205
[45]巫绪淘,杨伯源,数值外插法求解空间裂纹应力强度因子的研究,合肥工业大学学报(自然科学版),1999,22(4):26-31
[46] Way S., Pitting due to rolling contact, ASME Journal of Applied Mechanics, 1935, 2:49-58
[47]乐金朝,汤任基,双相材料空间中平片界面裂纹问题的超奇异积分-微分方程,科学通报,1996,41(15):1431～1433
[48]乐金朝,王博,刘文廷等,双相材料空间中受剪切载荷作用的平片裂纹问题的超奇异积分方程解法,工程力学增刊,1996,296-301
[49]乐金朝,汤任基,王复明等,双材料中平片裂纹问题的超奇异积分方程解法,应用力学学报,1999,16(4):1～6
[50]杨德全,赵忠生,边界元理论及应用,北京,北京理工大学出版社,2002
[51]秦太验,三维断裂力学的有限部积分-边界元法研究,上海交通大学博士论文,1993
[52]陈梦成,两相材料界面附近(含界面)三维断裂力学问题的超奇异积分方程方法,上海交通大学博士论文,1997
[53] Weaver J., Three-dimensional crack analysis, International Journal of Solids Structures, 1977, 13: 321-330
[54] Kassir M.K. , Stress-intensity factor for a three-dimensional rectangular crack, Transactions of ASME, 1981, 48: 309-312
[55] Nisitani. H. , Murakami Y. , Stress intensity factors of on elliptical crack of a semi-elliptical crack subjected to tension. Int. J. Fracture, 1974, 10: 353-368
[56]中国航空研究院,应力强度因子手册,北京,科学出版社,
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[2]尹双增,断裂力学,北京,清华大学出版社,1992
[3]沈成康,断裂力学,上海,同济大学出版社,1996
[4] Kassir M. K.. Sih G. S., Three-dimensional stress distribution around an elliptical crack under arbitrary loading., J. Appl. Mech., 1963, 33 (E38): 601-644
[5] Kassir M. K., Sih G. S., Three-dimensional crack problems, Sih G C: Mechanics of Fracture II, Noordboff, leyden, 1975
[6]王银邦,非对称载荷作用的外部圆形裂纹问题,应用数学和力学,2001,22(1):9～15
[7]陈梦成,陈振建,汤任基,椭圆平片裂纹前沿应力强度因子解析解的超奇异积分方程方法.固体力学学报,1999,20(4):331-334
[8] Fleming J. R. and Suh N P., Mechanism of crack propagation in delamination wear, Wear, 1977, 44: 39-56
[9]Olzak M., Stupnicki J., Wijcik R., Investigation of crack propagation during contact by a finite element method, Wear, 1991, 146: 29-240
[10] Olzak M., Stupnicki J., Wijcik R., Numerical analysis of 3D cracks propagation in rail-wheel contact zone, The Netherlands: June 1992, 385-393
[11] Dubourg M C, Kalker J J,, Crack behavior under rolling contact fatigue, The Netherlands: June 1992, 373-384
[12] Bogdanski S., Brown M W., Modelling the three dimensional behavior of shallow rolling contact fatigue cracks in rails[A], Proceeding of the 5th International Conference on Contact Mechanics and Wear of Whell/Rail System[C], Japan: July 25 to 28, 2000, 9-16
[13] Kabo E., Ekberg A., Fatigue initiation in railway wheels on the influence of defects[A], Proceeding of the 5th International Conference on Contact Mechanics and Wear of Whell/Rail System[C], Japan: July 25 to 28, 2000, 17-22
[14] 杨庆生,孙艳丰,裂纹扩展过程的数值模拟,铁道学报,1997,19(5):116～120
[15] Cruse T. A., Numerical solutions in three dimensional elastostantics, Int. J. Solids Structures, 1969, 5: 1259-1274
[16]杜庆华,姚振汉,边界积分方程.边界元法的基本理论及其在弹性力学方面的若干工程应用,固体力学学报,1982,3(1):1-22
[17]杜庆华,稽醒,芩章志等,边界积分方程-边界元法的力学基础也工程应用,北京, 高等教育出版社,1989
[18] Du Qinghua, Yao Zhenhan, Cen Zhangzhi, Some applications of boundary element methods, boundary element-finite element coupling techniques in elastoplastic stress analysis, Acta Mechanica Solids Sinice, 1990, 3 (3): 327-340
[19]童丽萍,张晓萍,用边界元分析工程中的裂纹,郑州大学学报,1998,30(4):15—20
[20]刘建秀,乐金朝,弹性半空间中平片裂纹问题的超奇异积分方程方法,机械强度,1999,21(3):208-211
[21]陶伟明,郭乙木,三维有限裂纹体分析,固体力学学报,2001,22(3):256—262
[22] Cai Guozhong, Zhang Kangda, Stress intensity factors for semi-elliptical surface cracks in plates and cylindrical pressure vessels using the hybrid boundary element method, Engineering Fracture Mechanics, 1995, 52(6): 1035-1054
[23]周里群,李玉平,用边界元求解二维弹性接触中问题的改进方法,湘潭大学自然科学学报,1998,2:90-100
[24]徐玉辰,肖宏,用于边界元法的改进数值积分技术,燕山大学学报,2001,25(4):325-327
[25]佟维,考虑网格自动划分求敏度的边界元法,大连铁道学院学报,1997,18(1):10～16
[26]姚振汉,段小华,尹欣等,边界元法软件及其在工程与教学中的应用,重庆建筑大学学报,2000,22(6):84-87
[27] Noda Nao-aki, Miyoshi Sinsuke, Variation of stress intensity factor and crack opening displacement of semi-elliptical surface crack, International Journal of Fracture, 1996, 75: 19-48
[28] Murakami Y., Nwnat-Nasser S., Growth and stability of interacting surface flaws of arbitrary shape, Engineering Fracture Mechanics, 1983, 17 (3);193 — 210
[29] Murakami Y., Analysis of stress intensity factors of modes 1, II, and III for inclined surface cracks of arbitrary shape, Engineering Fracture Mechanics, 1985, 22(1): 101-114
[30]石田诚,鹤 秀登,野口博司,三次元裂问题高精度新解析法(第1报,解析基础理论无限体问题応用),日本机械学会论文集(A编),1993,59(561)1270-1278
[31]野口博司,石田 诚,鹤 秀登,三次元裂问题高精度新解析法(第2报,任意形状,表面·内部裂持半无限体问题～応用),日本机械学会论文集(A编),1993,59(561):1279-1286
[32]村上敬宜,兼田楨宏,八塚裕彦,接触下表面裂解析,日本机械学会论文集(C编),1985,51(467):1603-1610
[33]兼田楨宏,八塚裕彦,村上敬宜,接触下表面裂解析(裂接力作用),日本机械学会论文集(C编),1985,51(470):2618～2624
[34] Murakami Y., Kaneta M., Yatsuzuka Y., Analysis of surface crack propagation in
lubricated rolling contact, ASLE Transactions, 1985, 28 (1): 60-68
[35] Kaneta M., Yatsuzuka Y., Murakami Y., Mechanism of surface crack growth in lubricatedrolling/sliding spherical contact, Transactions of the ASME, 1986, 53: 354-360
[36]村上敬宜,一同题破坏力学応用,日本机械学会论文集(A编) 1993,59(558):283～290
[37] Ioakimidis N.I. , A natural approach to the introduction of finite-part integral into crack problems of three-dimensional elasticity, Engineering Fracture Mechanics, 1982, 16: 669~673
[38] Ioakimidis N.I. , Application of finite-part integral to the singular integral equations of crack problems in plane and three-dimensional elasticity, Acta Mechanics, 1982, 45: 31-47
[39]汤任基,断裂力学中的两类奇异积分方程,上海交通大学学报,1990,24:36-45
[40]幸筱流,陈梦成,汤任基,椭圆类平片裂纹二维有限部积分方程的多项式数值方法,南吕大学学报(工科类),1999,21(1):45-50
[41] Sohn G.H., Hong G.S., Application of singular integral equations to embedded planarcrack problems in finite body in boundary element 2, Brebbia C.A. Springer-Verlog, 1985,8: 57-67
[42] Hayashi K., Abe H., Stress intensity factors for a semi-elliptical crack in the surface of asemi-infinite solid, International Journal of Fracture, 1980, 16 (3): 275-285
[43] Fujimoto K., ItoH., YamamotoT., Effect of cracks on the contact pressure distribution.,Tribology Transactions. Vol.35, 1992, PP.683-695
[44] Keer L M., Bryant M D., A pitting model for rolling contact fatigue, Journal ofLubrication Technology, April 1983, 105: 198-205
[45]巫绪淘,杨伯源,数值外插法求解空间裂纹应力强度因子的研究,合肥工业大学学报(自然科学版),1999,22(4):26-31
[46] Way S., Pitting due to rolling contact, ASME Journal of Applied Mechanics, 1935, 2:49-58
[47]乐金朝,汤任基,双相材料空间中平片界面裂纹问题的超奇异积分-微分方程,科学通报,1996,41(15):1431～1433
[48]乐金朝,王博,刘文廷等,双相材料空间中受剪切载荷作用的平片裂纹问题的超奇异积分方程解法,工程力学增刊,1996,296-301
[49]乐金朝,汤任基,王复明等,双材料中平片裂纹问题的超奇异积分方程解法,应用力学学报,1999,16(4):1～6
[50]杨德全,赵忠生,边界元理论及应用,北京,北京理工大学出版社,2002
[51]秦太验,三维断裂力学的有限部积分-边界元法研究,上海交通大学博士论文,1993
[52]陈梦成,两相材料界面附近(含界面)三维断裂力学问题的超奇异积分方程方法,上海交通大学博士论文,1997
[53] Weaver J., Three-dimensional crack analysis, International Journal of Solids Structures, 1977, 13: 321-330
[54] Kassir M.K. , Stress-intensity factor for a three-dimensional rectangular crack, Transactions of ASME, 1981, 48: 309-312
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